TY - JOUR

T1 - Pathological solutions to the Euler-Lagrange equation and existence/regularity of minimizers in one-dimensional variational problems

AU - Gratwick, Richard

AU - Sedipkov, Aidys

AU - Sychev, Mikhail

AU - Tersenov, Aris

PY - 2017/3/1

Y1 - 2017/3/1

N2 - In this paper, we prove that if L(x,u,v)∈C3(R3→R), Lvv>0 and L≥α|v|+β, α>0, then all problems (1), (2) admit solutions in the class W1,1[a,b], which are in fact C3-regular provided there are no pathological solutions to the Euler equation (5). Here u∈C3[c,d[ is called a pathological solution to equation (5) if the equation holds in [c,d[, |u˙(x)|→∞ as x→d, and ‖u‖C[c,d]<∞. We also prove that the lack of pathological solutions to the Euler equation results in the lack of the Lavrentiev phenomenon, see Theorem 9; no growth assumptions from below are required in this result.

AB - In this paper, we prove that if L(x,u,v)∈C3(R3→R), Lvv>0 and L≥α|v|+β, α>0, then all problems (1), (2) admit solutions in the class W1,1[a,b], which are in fact C3-regular provided there are no pathological solutions to the Euler equation (5). Here u∈C3[c,d[ is called a pathological solution to equation (5) if the equation holds in [c,d[, |u˙(x)|→∞ as x→d, and ‖u‖C[c,d]<∞. We also prove that the lack of pathological solutions to the Euler equation results in the lack of the Lavrentiev phenomenon, see Theorem 9; no growth assumptions from below are required in this result.

UR - http://www.scopus.com/inward/record.url?scp=85012899981&partnerID=8YFLogxK

U2 - 10.1016/j.crma.2017.01.020

DO - 10.1016/j.crma.2017.01.020

M3 - Article

AN - SCOPUS:85012899981

VL - 355

SP - 359

EP - 362

JO - Comptes Rendus Mathematique

JF - Comptes Rendus Mathematique

SN - 1631-073X

IS - 3

ER -